Trigonometry Examples

$4 \csc^{2} (θ) - 25 = 0$

Step 1

Add $25$ to both sides of the equation.

$4 \csc^{2} (θ) = 25$

Step 2

Divide each term in $4 \csc^{2} (θ) = 25$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 \csc^{2} (θ) = 25$ by $4$ .

$\frac{4 \csc^{2} (θ)}{4} = \frac{25}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 \csc^{2} (θ)}{4} = \frac{25}{4}$

Step 2.2.1.2

Divide $\csc^{2} (θ)$ by $1$ .

$\csc^{2} (θ) = \frac{25}{4}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\csc (θ) = \pm \sqrt{\frac{25}{4}}$

Step 4

Simplify $\pm \sqrt{\frac{25}{4}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{25}{4}}$ as $\frac{\sqrt{25}}{\sqrt{4}}$ .

$\csc (θ) = \pm \frac{\sqrt{25}}{\sqrt{4}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $25$ as $5^{2}$ .

$\csc (θ) = \pm \frac{\sqrt{5^{2}}}{\sqrt{4}}$

Step 4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\csc (θ) = \pm \frac{5}{\sqrt{4}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $4$ as $2^{2}$ .

$\csc (θ) = \pm \frac{5}{\sqrt{2^{2}}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\csc (θ) = \pm \frac{5}{2}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$\csc (θ) = \frac{5}{2}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$\csc (θ) = - \frac{5}{2}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$\csc (θ) = \frac{5}{2}, - \frac{5}{2}$

Step 6

Set up each of the solutions to solve for $θ$ .

$\csc (θ) = \frac{5}{2}$

$\csc (θ) = - \frac{5}{2}$

Step 7

Solve for $θ$ in $\csc (θ) = \frac{5}{2}$ .

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Step 7.1

Take the inverse cosecant of both sides of the equation to extract $θ$ from inside the cosecant.

$θ = arccsc (\frac{5}{2})$

Step 7.2

Simplify the right side.

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Step 7.2.1

Evaluate $arccsc (\frac{5}{2})$ .

$θ = 23.57817847$

Step 7.3

The cosecant function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$θ = 180 - 23.57817847$

Step 7.4

Subtract $23.57817847$ from $180$ .

$θ = 156.42182152$

Step 7.5

Find the period of $\csc (θ)$ .

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Step 7.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 7.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 7.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 7.5.4

Divide $360$ by $1$ .

$360$

Step 7.6

The period of the $\csc (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 23.57817847 + 360 n, 156.42182152 + 360 n$ , for any integer $n$

Step 8

Solve for $θ$ in $\csc (θ) = - \frac{5}{2}$ .

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Step 8.1

Take the inverse cosecant of both sides of the equation to extract $θ$ from inside the cosecant.

$θ = arccsc (- \frac{5}{2})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Evaluate $arccsc (- \frac{5}{2})$ .

$θ = - 23.57817847$

Step 8.3

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$θ = 360 + 23.57817847 + 180$

Step 8.4

Simplify the expression to find the second solution.

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Step 8.4.1

Subtract $360 °$ from $360 + 23.57817847 + 180 °$ .

$θ = 360 + 23.57817847 + 180 ° - 360 °$

Step 8.4.2

The resulting angle of $203.57817847 °$ is positive, less than $360 °$ , and coterminal with $360 + 23.57817847 + 180$ .

$θ = 203.57817847 °$

Step 8.5

Find the period of $\csc (θ)$ .

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Step 8.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 8.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 8.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 8.5.4

Divide $360$ by $1$ .

$360$

Step 8.6

Add $360$ to every negative angle to get positive angles.

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Step 8.6.1

Add $360$ to $- 23.57817847$ to find the positive angle.

$- 23.57817847 + 360$

Step 8.6.2

Subtract $23.57817847$ from $360$ .

$336.42182152$

Step 8.6.3

List the new angles.

$θ = 336.42182152$

Step 8.7

The period of the $\csc (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 203.57817847 + 360 n, 336.42182152 + 360 n$ , for any integer $n$

Step 9

List all of the solutions.

$θ = 23.57817847 + 360 n, 156.42182152 + 360 n, 203.57817847 + 360 n, 336.42182152 + 360 n$ , for any integer $n$

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate $23.57817847 + 360 n$ and $203.57817847 + 360 n$ to $23.57817847 + 180 n$ .

$θ = 23.57817847 + 180 n, 156.42182152 + 360 n, 336.42182152 + 360 n$ , for any integer $n$

Step 10.2

Consolidate $156.42182152 + 360 n$ and $336.42182152 + 360 n$ to $156.42182152 + 180 n$ .

$θ = 23.57817847 + 180 n, 156.42182152 + 180 n$ , for any integer $n$